Consider a 2D image scan-plane where the echographic apparatus reports the Doppler velocity at each point X of the plane VD(X). The velocity is given at each frame time of the acquisition, and at least one image is necessary. The point coordinates X can be expressed in general as the two Cartesian coordinates X=(x,y) or the polar coordinates X=(r,θ), where the radius r is the distance from the focus and θ is the sector angle, transversal to the radial direction. FIG. 1 shows the acquired Doppler velocity along a scan line.
The invention is directed toward the evaluation of the 2D velocity vector field V(X) in term of Cartesian components Vx(X), Vy(X), or polar components Vr(X), Vθ(X).
A few solutions to this problem were previously disclosed (Ohtsuki and Tanaka, 1991; Ohtsuki and Tanaka 1996; Uejima et al 2010; Garcia et al 2010; Ohtsuki and Tanaka, 1999a, 1999b, 2001). The methods employed in these publications are all based on the same general concept and are here summarized. Differences are remarked along with the descriptions.
They consider polar coordinates such that the radial velocity is simply Vr=−VD, and the transversal velocity Vθ has to be computed. To this aim, the tentative assumption of flow incompressibility on the plane is made, and the transversal coordinate is computed by using the planar continuity equation that in polar coordinates reads
                                                        ∂                              rV                r                                                    ∂              r                                +                                                    ∂                V                            ⁢                                                          ⁢              ϑ                                      ∂              ϑ                                      =        0                            (        1        )            
Given that the radial velocity is known by the Doppler measurement in the echographic sector ranging from θ1 to θ2, the first term in the continuity equation (1) can be computed and the radial velocity is then evaluated using (1) by integration
                                          V            ϑ                    ⁡                      (                          r              ,              ϑ                        )                          =                  -                                    ∫                              ϑ                0                            ϑ                        ⁢                                                                                ∂                    r                                    ⁢                                                                          ⁢                                      V                    r                                                                    ∂                  r                                            ⁢                                                          ⁢                              ⅆ                ϑ                                                                        (        2        )            where the integration can start from any position, here indicated by θ0, where the value of the transversal velocity is known.
The approach described above is often expressed making use of the streamfunction ψ(r, θ) that permits to define the velocities as
                                          V            r                    ⁡                      (                          r              ,              ϑ                        )                          =                                            1              r                        ⁢                                          ∂                ψ                                            ∂                ϑ                                      ⁢                                                  ⁢                                          V                ϑ                            ⁡                              (                                  r                  ,                  ϑ                                )                                              =                      -                                          ∂                ψ                                            ∂                r                                                                        (        3        )            and it is immediate to show that velocities defined in this way automatically satisfy the continuity equation (1). Using this approach the streamfunction can be evaluated from integration of the known radial velocity and then the transversal velocity from differentiation (3). This leads to exactly the same result (2). In synthesis, all these approaches suggest the use of the continuity equation (2) to compute the transversal velocity component from the radial, or Doppler, velocity. FIG. 2 shows this procedure of reconstruction of the transversal velocity.
Unfortunately, the integration (2) gives the solution apart from an undetermined function of the radial coordinate and of time. In other words, for example, the value of the transversal velocity on one end of the scan sector, at θ=θ1 can be imposed, while the value at the other end, at θ=θ2, follows from integration (2). Typically, either values needs to be imposed; for example to zero if the sector is wide enough, or to a predefined value of the bounding tissue. In fact, the direct use of (2) gives unrealistic values because incompressibility is not exact, due to the presence of a third cross-plane component, and the use of a first order integration (2) eventually provokes the accumulation of all the errors.
The methods mentioned above propose different solutions to this problem. The most popular solution, initially disclosed in (Ohtsuki and Tanaka, 1991) later reported and refined in (Ohtsuki and Tanaka 1996; Uejima et al 2010) suggests the separation of the Doppler velocity into two components and apply incompressibility on the so-called “vortex” component only, assuming that the other is responsible for cross-plane motion. Other solutions (Ohtsuki and Tanaka, 1999a, 1999b, 2001) introduce a series of sink-source points. The recent proposal (Garcia et al 2010) suggests the integration of (2) along the two directions, and takes the average of the two. In general, mathematically rigorously, the transversal velocity obtained by integration (2) method must be corrected with an arbitrary function that permits to satisfy the boundary conditions at the two ends of the sector. The easiest is that of using a linear correction, which is equivalent to subtracting a function of the radial distance and time, from the Doppler velocity.
Thus, these existing methods present two principal drawbacks:                The conceptual drawback that the continuity equation cannot be satisfied, therefore the solution Vθ(r, θ) is—in fact—defined up to an arbitrary function f(r,θ) that may completely alter the solution. The existing methods essentially suggest a, still arbitrary, choice to define the shape of such a function.        The methodological drawback that the solution is found by integration along each sector separately: the solution at one radial position is computed independently from the solution at another radial position. This is a consequence of using a formulation employing first order partial differential equation. The solutions typically present discontinuities along the radial position.        